As the title says, I want to make sure $(\mathbb{Z}/k\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{R}=0$ and to check if my reasoning is correct.
Obviously, $0\otimes 1=(0+0)\otimes 1=0\otimes 1+0\otimes 1$, so $0\otimes 1=0$.
On the other hand, consider a simple tensor $a\otimes x$. Since $ka=0$, it follows that $$a\otimes x=a\otimes(xk/k)=(ka)\otimes(x/k)=0\otimes(x/k)=(x/k)(0\otimes 1)=0.$$ Thus every simple tensor is zero, and therefore $(\mathbb{Z}/k\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{R}=0$